You are in charge of a generator that supplies power to three different cities. A giant magnet spins within a wire housing, which is connected to the first city. The positive terminal is just above the spinning magnet - at 12:00 so to speak. Alternating north and south poles of the magnet swing by this terminal, producing alternating current (AC). If we measure voltage and time in just the right way, the first city sees a voltage of sin(t). The second city is fed from a wire housing that is oriented at an angle to the first - at 4:00 so to speak. Its voltage is sin(t-120°). The third

Question

You are in charge of a generator that supplies power to three different cities.  A giant magnet spins within a wire housing, which is connected to the first city.  The positive terminal is just above the spinning magnet - at 12:00 so to speak.  Alternating north and south poles of the magnet swing by this terminal, producing alternating current (AC).  If we measure voltage and time in just the right way, the first city sees a voltage of sin(t).  The second city is fed from a wire housing that is oriented at an angle to the first - at 4:00 so to speak.  Its voltage is sin(t-120°).  The third

anonymous · Answer

The third city is connected to wires at 8:00 on the spinning magnet.  Its voltage is sin(t-240°).
The power consumed, and hence the work done by the generator, is proportional to the voltage squared.  So the first city is pulling power at a rate of sin(t) squared, sometimes written sin(t)2.  All three cities consume power at the following rate:

sin(t)2 + sin(t-120°)2 + sin(t-240°)2

Simplify this expression - the ans